Finite Group Databasebeta

← #2737058

Subgroups — #2737058

Order 512.

2736 conjugacy classes of subgroups.

The subgroup lattice has 775 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic, complemented
2C2×#2513C211yesmaximal, characteristic, complemented
3#17156C214yesmaximal, complemented
4#17156C214yesmaximal, complemented
5#17066C212yesmaximal, complemented
6#17066C212yesmaximal, complemented
7#17022C212yesmaximal, complemented
8#15243C211yesmaximal, characteristic, complemented
9#15236C212yesmaximal, complemented
10#15215C212yesmaximal, complemented
11#15215C212yesmaximal, complemented
12#15180C211yesmaximal, characteristic, complemented
13#15179C212yesmaximal, complemented
14#15123C211yesmaximal, characteristic
15#14623C212yesmaximal, complemented
16#13936C211yesmaximal, characteristic, complemented
17C4×#394C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×#121C2212yes—
23C22×C4×D4C2211yescharacteristic
24#2542—28no—
25#2513—28no—
26C2×#399C2214yes—
27C2×#397C2212yes—
28C2×#397C2212yes—
29C2×#394C2211yescharacteristic
30C2×#392C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#390C2211yescharacteristic
35C2×#390C2212yes—
36C2×#390C2211yescharacteristic
37C2×#390C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#388C2212yes—
40C2×#388C2212yes—
41C2×#384C2212yes—
42C2×#382C2211yescharacteristic
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48#1443C2214yes—
49#1442C2212yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2212yes—
54#1442C2211yescharacteristic
55#1441C2212yes—
56#1441C2214yes—
57#1441C2211yescharacteristic
58#1441C2212yes—
59#1440C2212yes—
60#1440C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2212yes—
71#1439C2212yes—
72#1439C2212yes—
73#1438C2214yes—
74#1438C2214yes—
75#1438C2214yes—
76#1438C2214yes—
77#1438C2214yes—
78#1438C2211yescharacteristic
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83#1434C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2312yes—
86C23×D4C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C23×D4C2314yes—
89C24×C4C2312yes—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—44no—
92C2×#124—44no—
93C2×#124—44no—
94C2×#124—44no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—44no—
98C2×C22⋊Q8—44no—
99C2×C22⋊Q8—44no—
100C2×C22⋊Q8—44no—
101C2×#121—44no—
102C2×#121—44no—
103C2×#121—44no—
104C2×C4×D4—24no—
105C2×C4×D4—44no—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140#399—216no—
141#397—28no—
142#397—28no—
143#394—24no—
144#392—28no—
145#391—44no—
146#391—44no—
147#391—44no—
148#390—24no—
149#390—24no—
150#390—24no—
151#390—28no—
152#389—24no—
153#388—28no—
154#388—28no—
155#384—28no—
156#382—24no—
157#380—44no—
158C4×C4⋊C4—24no—
159C4×C4⋊C4D418yescomplemented
160C4×C22⋊C4—44no—
161C4×C22⋊C4—44no—
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C25C2412yes—
189C25C2412yes—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×D4—28no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4#4114yes—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C4⋊Q8—416no—
237#124—432no—
238C22⋊Q8—432no—
239C22⋊Q8—432no—
240#121—432no—
241C4×D4—416no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D414yes—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—44no—
329C2×C42C2×D414yes—
330C2×C42C2×D418yes—
331#95—22no—
332#95—22no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—22no—
338#95—22no—
339#95—44no—
340#95—42no—
341#95—44no—
342#95—44no—
343#95—42no—
344#95—44no—
345#95—42no—
346#95—42no—
347#95—42no—
348#95—42no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95C2×D418yes—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—24no—
375C24—24no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—24no—
380C24—24no—
381C24—22no—
382C24C2×#4114yes—
383C24C2511yescharacteristic
384C2×Q8—48no—
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—48no—
390C2×Q8—48no—
391C2×D4—28no—
392C2×D4—48no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—48no—
396C2×D4—416no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—48no—
401C2×D4—48no—
402C2×D4—88no—
403C2×D4—88no—
404C2×D4—88no—
405C2×D4—88no—
406C2×D4—88no—
407C2×D4—88no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4C22×D414yes—
491C22×C4C22×D412yes—
492C22×C4C22×D414yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C2×#4112yes—
496C22×C4H(5,𝔽2)14yes—
497C22×C4H(5,𝔽2)12yes—
498C22×C4H(5,𝔽2)12yes—
499C22×C4H(5,𝔽2)14yes—
500C22×C42−1+412yes—
501C22×C42−1+412yes—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—416no—
507C4⋊C4—416no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—48no—
513C4⋊C4—84no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—416no—
518C22⋊C4—416no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—416no—
523C22⋊C4—416no—
524C22⋊C4—416no—
525C22⋊C4—48no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C22⋊C4—88no—
531C42—28no—
532C42—48no—
533C42—44no—
534C42—44no—
535C42—48no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—44no—
579C23—42no—
580C23—44no—
581C23—42no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—42no—
589C23—42no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23C23×D411yescharacteristic
595C23C22×#4111yescharacteristic
596C23C2×H(5,𝔽2)12yes—
597C23C2×H(5,𝔽2)11yescharacteristic
598C23C2×H(5,𝔽2)12yes—
599C23C2×H(5,𝔽2)11yescharacteristic
600C23C2×2−1+411yescharacteristic
601C23C2×2−1+411yescharacteristic
602C23#58512yes—
603C23#58511yescharacteristic
604C23#58512yes—
605Q8—816no—
606D4—416no—
607D4—832no—
608D4—88no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—44no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—44no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—816no—
691C2×C4#53514yes—
692C2×C4#53514yes—
693C2×C4#53514yes—
694C2×C4#54814yes—
695C2×C4#56018yes—
696C2×C4#56018yes—
697C22—44no—
698C22—44no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—44no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—84no—
715C22—84no—
716C22—84no—
717C22—82no—
718C22—84no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22C2×#53511yescharacteristic
724C22C2×#53511yescharacteristic
725C22C2×#53511yescharacteristic
726C22C2×#54811yescharacteristic
727C22#348511yescharacteristic
728C22#348911yescharacteristic
729C22#350011yescharacteristic
730C22#350512yes—
731C22#351412yes—
732C22#351411yescharacteristic
733C22#352011yescharacteristic
734C22C2×#56012yes—
735C22C2×#56012yes—
736C22#353912yes—
737C22#354012yes—
738C22#354112yes—
739C22#354712yes—
740C22#354711yescharacteristic
741C22#355211yescharacteristic
742C22#355412yes—
743C22#355412yes—
744C22#356212yes—
745C22#356212yes—
746C4—24no—
747C4—44no—
748C4—44no—
749C4—42no—
750C4—42no—
751C4—82no—
752C4—81no—
753C4—82no—
754C4—82no—
755C4—82no—
756C4—84no—
757C4—84no—
758C4—81no—
759C4—81no—
760C4#281818yes—
761C2—84no—
762C2—162no—
763C2C2×#281811yescharacteristic
764C2#3912411yescharacteristic
765C2#3917111yescharacteristic
766C2#3972211yescharacteristic
767C2#4265211yescharacteristic
768C2#4268311yescharacteristic
769C2#4299511yescharacteristic
770C2#4561612yes—
771C2#4569712yes—
772C2#5304712yes—
773C2#5305112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#15123C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#2542C2semidirect
2. C2×#2513C2semidirect
3. #17156C2semidirect
4. #17156C2semidirect
5. #17066C2semidirect
6. #17066C2semidirect
7. #17022C2semidirect
8. #15243C2semidirect
9. #15236C2semidirect
10. #15215C2semidirect
11. #15215C2semidirect
12. #15180C2semidirect
13. #15179C2semidirect
15. #14623C2semidirect
16. #13936C2semidirect
17. C4×#394C2semidirect
159. C4×C4⋊C4D4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#25422
C2×#25132
#171562
#170662
#170222
#152432
#152362
#152152
#151802
#151792
#151232
#146232
#139362
C4×#3942